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MEHRAN KARDAR: OK, let's start.
00:00:25.160 --> 00:00:31.730
So in this class,
we focused mostly
00:00:31.730 --> 00:00:41.150
on having some slab of material
and having some configuration
00:00:41.150 --> 00:00:45.310
of some kind of a field inside.
00:00:45.310 --> 00:00:49.720
And we said that,
basically, we are
00:00:49.720 --> 00:00:52.810
going to be interested
close to, let's say,
00:00:52.810 --> 00:00:56.950
phase transition and some
quantity that changes
00:00:56.950 --> 00:00:57.950
at the phase transition.
00:00:57.950 --> 00:01:01.330
We are interested in figuring
out the singularities
00:01:01.330 --> 00:01:03.350
associated with that.
00:01:03.350 --> 00:01:06.710
And we can coarse grain.
00:01:06.710 --> 00:01:12.470
Once we have coarse grained, we
have the field m, potentially
00:01:12.470 --> 00:01:16.950
a vector, that is characterized
throughout this material.
00:01:16.950 --> 00:01:20.960
So it's a field
that's function of x.
00:01:20.960 --> 00:01:28.780
And by integrating out over
a lot of degrees of freedom,
00:01:28.780 --> 00:01:32.170
we can focus on the
probability of finding
00:01:32.170 --> 00:01:37.080
different configurations
of this field.
00:01:37.080 --> 00:01:40.150
And this probability
we constructed
00:01:40.150 --> 00:01:46.010
on the basis of a number
of simple assumptions
00:01:46.010 --> 00:01:50.780
such as locality, which
implied that we would write
00:01:50.780 --> 00:01:56.360
this probability as a
product of contributions
00:01:56.360 --> 00:02:01.590
of different parts, which in the
exponent becomes an integral.
00:02:01.590 --> 00:02:05.520
And then we would put
within this all kinds
00:02:05.520 --> 00:02:09.560
of things that are consistent
with the symmetries
00:02:09.560 --> 00:02:10.599
of the problem.
00:02:10.599 --> 00:02:13.370
So if, for example,
this is a field that
00:02:13.370 --> 00:02:16.090
is invariant on
the rotations, we
00:02:16.090 --> 00:02:22.735
would be having terms such as
m squared, m to the fourth,
00:02:22.735 --> 00:02:23.580
and so forth.
00:02:26.630 --> 00:02:28.620
But the interesting thing
was that, of course,
00:02:28.620 --> 00:02:32.380
there is some interaction
with the neighborhoods.
00:02:32.380 --> 00:02:36.520
Those neighborhood interactions
we can, in the continuum limit,
00:02:36.520 --> 00:02:40.670
inclement by putting terms that
are proportional to radiant
00:02:40.670 --> 00:02:42.540
of m and so forth.
00:02:50.240 --> 00:02:52.280
So there is a lot
of things that you
00:02:52.280 --> 00:02:55.620
could put consistent
with symmetry
00:02:55.620 --> 00:02:58.550
and presumably be as
general as possible.
00:02:58.550 --> 00:03:00.880
You could have
this and the terms
00:03:00.880 --> 00:03:04.620
in this coefficients
of this expansion would
00:03:04.620 --> 00:03:08.260
be these phenomenological
parameters characterizing
00:03:08.260 --> 00:03:11.230
this probability,
function of all kinds
00:03:11.230 --> 00:03:13.090
of microscopic
degrees of freedom,
00:03:13.090 --> 00:03:15.310
as well as microscopic
constraints
00:03:15.310 --> 00:03:19.140
such as temperature,
pressures, et cetera.
00:03:19.140 --> 00:03:27.390
Now the question that we have
is if I start with a system,
00:03:27.390 --> 00:03:30.190
let's say a configuration
where everybody's pointing up
00:03:30.190 --> 00:03:33.990
or some other configuration
that is not the equilibrium
00:03:33.990 --> 00:03:36.930
configuration, how
does the probability
00:03:36.930 --> 00:03:41.290
evolve to become
something like this?
00:03:41.290 --> 00:03:44.970
Now we are interested
therefore in m
00:03:44.970 --> 00:03:49.030
that is a function
of position and time.
00:03:49.030 --> 00:03:54.180
And since I want to use t for
time, this coefficient of m
00:03:54.180 --> 00:03:57.366
squared that we were
previously calling t,
00:03:57.366 --> 00:04:02.250
I will indicate by r, OK?
00:04:02.250 --> 00:04:04.470
There are various types
of dynamics that you
00:04:04.470 --> 00:04:07.020
can look at for this problem.
00:04:07.020 --> 00:04:09.562
I will look at the class
that is dissipative.
00:04:15.880 --> 00:04:20.100
And its inspiration
is the Brownian motion
00:04:20.100 --> 00:04:22.980
that we discussed
last time where
00:04:22.980 --> 00:04:27.850
we saw that when you put
a particle in a fluid
00:04:27.850 --> 00:04:31.910
to a very good approximation
when the fluid is viscous,
00:04:31.910 --> 00:04:34.880
it is the velocity that is
proportional to the force
00:04:34.880 --> 00:04:38.600
and you can't ignore inertial
effects such as mass times
00:04:38.600 --> 00:04:41.330
acceleration and you
write an equation
00:04:41.330 --> 00:04:45.930
that is linear in position.
00:04:45.930 --> 00:04:48.210
Velocity is the
linear derivative
00:04:48.210 --> 00:04:50.180
of position, which
is the variable that
00:04:50.180 --> 00:04:52.130
is of interest to you.
00:04:52.130 --> 00:04:55.000
So here the variable that
is of interest to us,
00:04:55.000 --> 00:04:59.010
is this magnetization that is
changing as a function of time.
00:04:59.010 --> 00:05:02.100
And the equation
that we write down
00:05:02.100 --> 00:05:09.400
is the derivative of this
field with respect to time.
00:05:09.400 --> 00:05:14.650
And again, for the
Brownian motion,
00:05:14.650 --> 00:05:18.400
the velocity was
proportional to the force.
00:05:18.400 --> 00:05:22.380
The constant of proportionality
was some kind of a mobility
00:05:22.380 --> 00:05:24.340
that you would
have in the fluid.
00:05:24.340 --> 00:05:27.530
So continuing with
that inspiration,
00:05:27.530 --> 00:05:29.740
let's put some kind
of a mobility here.
00:05:29.740 --> 00:05:32.830
I should really put a
vector field here, but just
00:05:32.830 --> 00:05:38.420
for convenience, let's
just focus on one component
00:05:38.420 --> 00:05:40.830
and see what happens.
00:05:40.830 --> 00:05:44.730
Now what is the force?
00:05:44.730 --> 00:05:49.670
Presumably, each
location individually
00:05:49.670 --> 00:05:52.570
feels some kind of a force.
00:05:52.570 --> 00:05:56.660
And typically when we had the
Brownian particle, the force
00:05:56.660 --> 00:05:58.420
we were getting
from the derivative
00:05:58.420 --> 00:06:02.125
of the potential with
respect to the [? aviation ?]
00:06:02.125 --> 00:06:06.390
of the position, the field
that we are interested.
00:06:06.390 --> 00:06:13.030
So the analog of our
potential is the energy
00:06:13.030 --> 00:06:17.280
that we have over here,
and in the same sense
00:06:17.280 --> 00:06:21.430
that we want this effective
potential, if you like,
00:06:21.430 --> 00:06:24.760
to govern the
equilibrium behavior--
00:06:24.760 --> 00:06:28.050
and again, recall that for the
case of the Brownian particle,
00:06:28.050 --> 00:06:31.670
eventually the probability was
related to the potential by e
00:06:31.670 --> 00:06:36.450
to the minus beta v. This
is the analog of beta v, now
00:06:36.450 --> 00:06:38.770
considered in the entire field.
00:06:38.770 --> 00:06:45.740
So the analog of the force is
a derivative of this beta H,
00:06:45.740 --> 00:06:53.600
so this would be our beta H
with respect to the variable
00:06:53.600 --> 00:06:55.590
that I'm trying to change.
00:06:58.930 --> 00:07:05.325
And since I don't have just
one variable, but a field,
00:07:05.325 --> 00:07:08.350
the analog of the derivative
becomes this functional
00:07:08.350 --> 00:07:09.980
derivative.
00:07:09.980 --> 00:07:14.160
And in the same sense that
the Brownian particle,
00:07:14.160 --> 00:07:17.540
the Brownian motion,
is trying to pull you
00:07:17.540 --> 00:07:20.470
towards the minimum
of the potential,
00:07:20.470 --> 00:07:24.450
this is an equation that,
if I kind of stop over here,
00:07:24.450 --> 00:07:28.090
tries to put the particle
towards the minimum
00:07:28.090 --> 00:07:33.000
of this beta H.
00:07:33.000 --> 00:07:36.530
Now the reason that the Brownian
particle didn't go and stick
00:07:36.530 --> 00:07:40.220
at one position, which was
the minimum, but fluctuated,
00:07:40.220 --> 00:07:43.310
was of course, that we
added the random force.
00:07:43.310 --> 00:07:46.970
So there is some kind of an
analog of a random force that
00:07:46.970 --> 00:07:51.157
we can put either in front of
[? mu ?] or imagine that if we
00:07:51.157 --> 00:07:56.100
added it to [? mu, ?]
and we put it over here.
00:07:56.100 --> 00:07:59.740
Now for the case of
the Brownian particle,
00:07:59.740 --> 00:08:04.930
the assumption was that if
I evaluate eta at some time,
00:08:04.930 --> 00:08:08.860
and eta at another
time t2 and t1,
00:08:08.860 --> 00:08:17.520
that this was related to 2D
delta function t1 minus t2.
00:08:17.520 --> 00:08:23.840
Now of course here, at each
location, I have a noise term.
00:08:23.840 --> 00:08:28.400
So this noise carries
an index, which
00:08:28.400 --> 00:08:31.980
indicates the position, which
is, of course, a vector in D
00:08:31.980 --> 00:08:34.309
dimension in principal.
00:08:34.309 --> 00:08:38.610
And there is no reason to
imagine that the noise here
00:08:38.610 --> 00:08:42.210
that comes from all kinds of
microscopic degrees of freedom
00:08:42.210 --> 00:08:45.310
that we have
integrated out should
00:08:45.310 --> 00:08:48.650
have correlation with the
noise at some other point.
00:08:48.650 --> 00:08:54.520
So the simplest assumption is
to also put the delta function
00:08:54.520 --> 00:08:58.710
in the positions, OK?
00:08:58.710 --> 00:09:02.340
So if I take this beta
H that I have over there
00:09:02.340 --> 00:09:05.530
and take the functional
derivative, what do I get?
00:09:05.530 --> 00:09:17.110
I will get that the m of xt by
dt is the function of-- oops,
00:09:17.110 --> 00:09:20.650
and I forgot to put a
the minus sign here.
00:09:20.650 --> 00:09:23.290
The force is minus
the potential,
00:09:23.290 --> 00:09:26.100
and this is basically
going down the gradient,
00:09:26.100 --> 00:09:28.060
so I need to put that.
00:09:28.060 --> 00:09:31.530
So I have to take a
derivative of this.
00:09:31.530 --> 00:09:33.765
First of all I can take
a derivative with respect
00:09:33.765 --> 00:09:35.780
to m itself.
00:09:35.780 --> 00:09:40.055
I will get minus rm.
00:09:44.640 --> 00:09:48.840
Well, actually, let's
put the minus out front.
00:09:48.840 --> 00:09:52.580
And then I will get
the derivative of m
00:09:52.580 --> 00:09:56.280
to the fourth, which is
[? 4umq, ?] all kinds of terms
00:09:56.280 --> 00:09:57.790
like this.
00:09:57.790 --> 00:10:01.110
Then the terms that come
from the derivatives
00:10:01.110 --> 00:10:05.650
and taking the derivative
with respect to gradient
00:10:05.650 --> 00:10:08.400
will give me K
times the gradient.
00:10:08.400 --> 00:10:10.780
But then in the
functional derivative,
00:10:10.780 --> 00:10:13.370
I have to take
another derivative,
00:10:13.370 --> 00:10:17.320
converting this to minus
K log Gaussian of m.
00:10:17.320 --> 00:10:20.050
And then I would have
L the fourth derivative
00:10:20.050 --> 00:10:24.310
from the next term and so forth.
00:10:24.310 --> 00:10:26.330
And on top of
everything else, I will
00:10:26.330 --> 00:10:31.080
have the noise who's statistics
I have indicated above.
00:10:34.650 --> 00:10:37.880
So this entity,
this equation, came
00:10:37.880 --> 00:10:42.820
from the Landau-Ginzburg
model, and it
00:10:42.820 --> 00:10:51.325
is called a time dependent
Landau-Ginzburg equation.
00:10:54.540 --> 00:11:02.120
And so would be the analog of
the Brownian type of equation,
00:11:02.120 --> 00:11:05.810
now for an entire field.
00:11:05.810 --> 00:11:08.820
Now we're going to have
a lot of difficulty
00:11:08.820 --> 00:11:12.750
in the short amount of time
that is left to us to deal
00:11:12.750 --> 00:11:15.320
with this nonlinear
equation, so we
00:11:15.320 --> 00:11:18.160
are going to do the same
thing that we did in order
00:11:18.160 --> 00:11:22.300
to capture the properties
of the Landau-Ginzburg model
00:11:22.300 --> 00:11:23.720
qualitatively.
00:11:23.720 --> 00:11:26.410
Which is to ignore
nonlinearities,
00:11:26.410 --> 00:11:30.000
so it's a kind of Gaussian
version of the model.
00:11:30.000 --> 00:11:33.810
So what we did
was we linearized.
00:11:33.810 --> 00:11:34.720
AUDIENCE: Question.
00:11:34.720 --> 00:11:35.960
MEHRAN KARDAR: Yes?
00:11:35.960 --> 00:11:38.690
AUDIENCE: Why does the
sign in front of the K term
00:11:38.690 --> 00:11:40.830
change relative to the others?
00:11:40.830 --> 00:11:48.140
MEHRAN KARDAR: OK, so when you
have the function 0 of a field
00:11:48.140 --> 00:11:56.010
gradient, et
cetera, you can show
00:11:56.010 --> 00:12:01.510
that the functional
derivative, the first term
00:12:01.510 --> 00:12:06.940
is the ordinary
type of derivative.
00:12:06.940 --> 00:12:12.152
And then if you think about the
variations that are carrying m
00:12:12.152 --> 00:12:17.870
and with the gradient and you
write it as m plus delta m
00:12:17.870 --> 00:12:21.660
and then make sure that you
take the delta m outside,
00:12:21.660 --> 00:12:26.280
you need to do an integration
by part that changes the sign.
00:12:26.280 --> 00:12:28.566
And the next term
would be the gradient
00:12:28.566 --> 00:12:32.760
of the phi by the [? grad ?]
m, and the next term
00:12:32.760 --> 00:12:35.550
would be log [INAUDIBLE]
of [INAUDIBLE] gradient
00:12:35.550 --> 00:12:38.176
of m squared and so forth.
00:12:38.176 --> 00:12:43.436
It alternates and so by
explicitly calculating
00:12:43.436 --> 00:12:45.580
the difference of
two functionals
00:12:45.580 --> 00:12:51.320
with this integration evaluated
at m and m plus delta m
00:12:51.320 --> 00:12:55.102
and pulling out everything
that is proportional to delta m
00:12:55.102 --> 00:12:56.910
you can prove these expressions.
00:13:00.920 --> 00:13:05.880
So we linearize this equation,
which I've already done that.
00:13:05.880 --> 00:13:09.980
I cross out the mq term and
any other nonlinear terms,
00:13:09.980 --> 00:13:12.630
so I only keep the linear term.
00:13:12.630 --> 00:13:16.065
And then I do a
Fourier transform.
00:13:21.330 --> 00:13:27.660
So basically I switch from
the position representation
00:13:27.660 --> 00:13:32.620
to the Fourier transform that's
called m tilde of q, I think.
00:13:32.620 --> 00:13:35.770
x is replaced with q.
00:13:35.770 --> 00:13:42.220
And then the equation
for the field m linears.
00:13:42.220 --> 00:13:46.900
The form separates out
into independent equations
00:13:46.900 --> 00:13:51.660
for the components that
are characterized by q.
00:13:51.660 --> 00:13:55.200
So the Fourier transform of the
left hand side is just this.
00:13:55.200 --> 00:13:58.460
Fourier transform of
the right hand side
00:13:58.460 --> 00:14:05.660
will give me minus mu
r plus K q squared.
00:14:05.660 --> 00:14:10.140
The derivative of [INAUDIBLE]
will give me a minus q squared.
00:14:10.140 --> 00:14:15.290
The derivative of the next term
L q to the forth, et cetera.
00:14:15.290 --> 00:14:17.776
And I've only kept
the linear term,
00:14:17.776 --> 00:14:21.510
so I have m tilde of q and t.
00:14:21.510 --> 00:14:23.580
And then I have the
Fourier transforms
00:14:23.580 --> 00:14:27.000
of the noise eta of x and t.
00:14:27.000 --> 00:14:29.820
They [? call it ?] eta
tilde of q [? m. ?]
00:14:33.590 --> 00:14:39.870
So you can see that
each mode satisfy
00:14:39.870 --> 00:14:42.940
a separate linear equation.
00:14:45.570 --> 00:14:49.290
So this equation is
actually very easy
00:14:49.290 --> 00:14:56.960
to solve for any linear
equation m tilde of q and t.
00:14:56.960 --> 00:15:00.230
If I didn't have the
noise, so I would
00:15:00.230 --> 00:15:05.360
start with some value
at t [? close ?] to 0,
00:15:05.360 --> 00:15:09.940
and that value would
decay exponentially
00:15:09.940 --> 00:15:14.660
in the characteristic time
that I will call tao of q.
00:15:14.660 --> 00:15:26.980
And 1 over tao of q is
simply this mu r plus K q2
00:15:26.980 --> 00:15:30.860
and so forth.
00:15:30.860 --> 00:15:34.920
Now once you have noise,
essentially each one
00:15:34.920 --> 00:15:39.180
of these noises acts like
an initial condition.
00:15:39.180 --> 00:15:43.300
And so the full
answer is an integral
00:15:43.300 --> 00:15:48.470
over all of these noises from
0 tilde time t of interest.
00:15:48.470 --> 00:15:54.260
Dt prime the noise that
occurs at a time t prime,
00:15:54.260 --> 00:15:57.515
and the noise that occurs at
time t [? trime ?] relaxes
00:15:57.515 --> 00:16:06.350
with this into the minus
t minus t prime tao of q.
00:16:06.350 --> 00:16:12.420
So that's the solution to
that linear noisy equations,
00:16:12.420 --> 00:16:14.320
basically [? sequencing ?]
[INAUDIBLE].
00:16:17.970 --> 00:16:24.330
So one of the things
that we now see
00:16:24.330 --> 00:16:33.630
is that essentially the
different Fourier components
00:16:33.630 --> 00:16:35.080
of the field.
00:16:35.080 --> 00:16:38.150
Each one of them
is independently
00:16:38.150 --> 00:16:41.710
relaxing to something,
and each one of them
00:16:41.710 --> 00:16:46.690
has a characteristic
relaxation time.
00:16:46.690 --> 00:16:54.360
As I go towards smaller
and smaller values of q,
00:16:54.360 --> 00:16:59.670
this rate becomes smaller,
and the relaxation time
00:16:59.670 --> 00:17:01.580
becomes larger.
00:17:01.580 --> 00:17:03.880
So essentially shortwave
wavelength modes
00:17:03.880 --> 00:17:07.630
that correspond to large
q, they relax first.
00:17:07.630 --> 00:17:11.280
Longer wavelength modes
will relax later on.
00:17:11.280 --> 00:17:19.480
And you can see that the largest
relaxation time, tao max,
00:17:19.480 --> 00:17:25.243
corresponds to q equals to 0
is simply 1 over [? nu ?] r.
00:17:28.349 --> 00:17:37.270
So I can pluck this to a
max as a function of r,
00:17:37.270 --> 00:17:41.200
and again in this theory,
the Gaussian theory,
00:17:41.200 --> 00:17:45.410
we saw only makes sense as
long as all is positive.
00:17:45.410 --> 00:17:49.410
So I have to look only
on the positive axis.
00:17:49.410 --> 00:17:53.060
And I find that
the relaxation time
00:17:53.060 --> 00:17:57.060
for the entire system for
the longest wavelength
00:17:57.060 --> 00:18:02.310
actually diverges
as r goes to 0.
00:18:02.310 --> 00:18:05.560
And recall that r,
in our perspective,
00:18:05.560 --> 00:18:09.320
is really something that is
proportional to T minus Tc.
00:18:12.260 --> 00:18:16.800
So basically we find that
as we are approaching
00:18:16.800 --> 00:18:20.810
the critical point,
the time it takes
00:18:20.810 --> 00:18:23.850
for the entirety of the
system or the longest
00:18:23.850 --> 00:18:30.820
wavelength [? modes ?] to relax
diverges as 1 over T minus Tc.
00:18:30.820 --> 00:18:35.010
There's an exponent that
shows this, so called,
00:18:35.010 --> 00:18:36.660
critical slowing down.
00:18:43.950 --> 00:18:46.400
Yes?
00:18:46.400 --> 00:18:48.200
AUDIENCE: In principal,
why couldn't you
00:18:48.200 --> 00:18:53.930
have that r becomes negative
if you restrict your range of q
00:18:53.930 --> 00:18:58.930
to be outside of some value
and not go arbitrarily
00:18:58.930 --> 00:19:01.380
close to the origin.
00:19:01.380 --> 00:19:05.230
MEHRAN KARDAR: You could, but
what's the physics of that?
00:19:05.230 --> 00:19:06.056
See.
00:19:06.056 --> 00:19:08.550
AUDIENCE: I'm wondering if
there is or is that not needed.
00:19:08.550 --> 00:19:09.300
MEHRAN KARDAR: No.
00:19:09.300 --> 00:19:14.360
OK, so the physics of that could
be that you have a system that
00:19:14.360 --> 00:19:21.950
has some finite size, [? l. ?]
Then the largest q that you
00:19:21.950 --> 00:19:27.030
could have [? or ?] the smallest
q that you could have would be
00:19:27.030 --> 00:19:31.680
the [? 1/l. ?] So in principal,
for that you can go slightly
00:19:31.680 --> 00:19:32.610
negative.
00:19:32.610 --> 00:19:37.180
You still cannot go too negative
because ultimately this will
00:19:37.180 --> 00:19:39.490
overcome that.
00:19:39.490 --> 00:19:42.020
But again, we are
interested in singularities
00:19:42.020 --> 00:19:46.884
that we kind of know arise
in the limit of [INAUDIBLE].
00:19:52.850 --> 00:19:56.150
I also recall, there is a
time that we see as diverging
00:19:56.150 --> 00:19:57.810
as r goes to 0.
00:19:57.810 --> 00:20:02.060
Of course, we identified
before a correlation length
00:20:02.060 --> 00:20:06.380
from balancing these two terms,
and the correlation length
00:20:06.380 --> 00:20:10.540
is square root of K over r,
which is again proportional
00:20:10.540 --> 00:20:15.500
to T minus Tc and diverges
with a square root singularity.
00:20:15.500 --> 00:20:23.640
So we can see that this
tao max is actually related
00:20:23.640 --> 00:20:30.622
to the psi squared over Nu K.
00:20:30.622 --> 00:20:34.150
And it is also
related towards this.
00:20:34.150 --> 00:20:47.610
We can see that our tao of
q, basically, if q is large
00:20:47.610 --> 00:20:54.980
such that qc is larger than
1 and the characteristic time
00:20:54.980 --> 00:21:03.340
is going to be 1 over [? nu ?] K
times the inverse of q squared.
00:21:03.340 --> 00:21:07.390
And the inverse of q is
something like a wavelength.
00:21:07.390 --> 00:21:12.290
Whereas, ultimately, this
saturates for qc what is less
00:21:12.290 --> 00:21:18.440
than 1 [? long ?] wavelengths
to c squared over [? nu ?] K.
00:21:18.440 --> 00:21:26.230
So basically you see
things at very short range,
00:21:26.230 --> 00:21:29.040
at length scales that are
much less than the correlation
00:21:29.040 --> 00:21:33.760
length of the system, that
the characteristic time will
00:21:33.760 --> 00:21:37.475
depend on the length scale that
you are looking at squared.
00:21:41.130 --> 00:21:48.070
Now you have seen times
scaling as length squared
00:21:48.070 --> 00:21:51.870
from diffusion, so
essentially this
00:21:51.870 --> 00:21:56.256
is some kind of a
manifestation of diffusion,
00:21:56.256 --> 00:22:01.620
but as you perturb the system,
let's say at short distances,
00:22:01.620 --> 00:22:03.930
there's some equilibrium system.
00:22:03.930 --> 00:22:07.120
Let's say we do some
perturbation to it
00:22:07.120 --> 00:22:09.790
at some point, and
that perturbation
00:22:09.790 --> 00:22:14.940
will start to expand
diffusively until it reaches
00:22:14.940 --> 00:22:18.050
the size of the correlation
length, at which point
00:22:18.050 --> 00:22:21.050
it stops because essentially
correlation length is
00:22:21.050 --> 00:22:25.390
an individual block that doesn't
know about individual blocks,
00:22:25.390 --> 00:22:26.900
so the influence does not last.
00:22:30.480 --> 00:22:33.380
So quite generally,
what you find--
00:22:33.380 --> 00:22:36.480
so we solved the
linearized version
00:22:36.480 --> 00:22:38.450
of the Landau-Ginzburg
model, but we
00:22:38.450 --> 00:22:43.960
know that, say, the critical
behaviors for the divergence
00:22:43.960 --> 00:22:47.040
of the correlation length
that is predicted here
00:22:47.040 --> 00:22:49.030
is not correct in
three dimensions,
00:22:49.030 --> 00:22:50.830
things get modified.
00:22:50.830 --> 00:22:54.580
So these kind of exponents
that come from diffusion also
00:22:54.580 --> 00:22:56.690
gets modified.
00:22:56.690 --> 00:23:01.640
And quite generally, you
find that the relaxation time
00:23:01.640 --> 00:23:09.610
of a mode of wavelength q
is going to behave something
00:23:09.610 --> 00:23:21.140
like wavelength,
which is 1 over q,
00:23:21.140 --> 00:23:27.710
rather than squared
as some exponent z.
00:23:27.710 --> 00:23:33.190
And then there is some
function of the product
00:23:33.190 --> 00:23:34.990
of the wavelength
you are looking at
00:23:34.990 --> 00:23:39.020
and the correlation length
so that you will cross over
00:23:39.020 --> 00:23:42.260
from one behavior
to another behavior
00:23:42.260 --> 00:23:44.280
as you are looking
at length scales that
00:23:44.280 --> 00:23:46.050
are smaller than the
correlation length
00:23:46.050 --> 00:23:49.090
or larger than the
correlation length.
00:23:49.090 --> 00:23:52.600
And to get what
this exponent z is,
00:23:52.600 --> 00:23:56.850
you have to do study
of the nonlinear model
00:23:56.850 --> 00:23:58.950
in the same sense
that, in order to get
00:23:58.950 --> 00:24:01.350
the correction to the
exponent [? nu ?],
00:24:01.350 --> 00:24:03.500
we had to do epsilon expansion.
00:24:03.500 --> 00:24:05.115
You have to do
something similar,
00:24:05.115 --> 00:24:10.970
and you'll fine that
at higher changes,
00:24:10.970 --> 00:24:15.240
it goes like some correction
that does not actually
00:24:15.240 --> 00:24:18.580
start at order of epsilon but
at order of epsilon squared.
00:24:18.580 --> 00:24:24.520
But there's essentially
some modification
00:24:24.520 --> 00:24:27.920
of the qualitative
behavior that we
00:24:27.920 --> 00:24:33.180
can ascribe to the fusion
of independent modes
00:24:33.180 --> 00:24:37.190
exists quite generally
and universal exponents
00:24:37.190 --> 00:24:40.700
different from [? to ?]
will emerge from that.
00:24:45.600 --> 00:24:51.900
Now it turns out that this
is not the end of the story
00:24:51.900 --> 00:24:54.820
because we have seen
that the same probability
00:24:54.820 --> 00:25:00.650
distribution can describe
a lot of different systems.
00:25:00.650 --> 00:25:04.910
Let's say the focus on
the case of n equals to 1.
00:25:08.680 --> 00:25:11.760
So then this Landau-Ginzburg
that I described
00:25:11.760 --> 00:25:15.680
for you can describe,
let's say, the Ising model,
00:25:15.680 --> 00:25:24.040
which describes
magnetizations that lie
00:25:24.040 --> 00:25:26.700
along the particular direction.
00:25:26.700 --> 00:25:31.040
So that it can also describe
liquid gas phenomena
00:25:31.040 --> 00:25:34.480
where the order parameter is
the difference in density,
00:25:34.480 --> 00:25:37.760
if you like, between
the liquid and the gas.
00:25:37.760 --> 00:25:40.970
Yet another example
that it describes
00:25:40.970 --> 00:25:43.650
is the mixing of an alloy.
00:25:50.110 --> 00:25:58.380
So let's, for example, imagine
brass that has a composition
00:25:58.380 --> 00:26:01.550
x that goes between 0 and 1.
00:26:01.550 --> 00:26:04.345
On one end, let's say
you have entirely copper
00:26:04.345 --> 00:26:08.250
and on the other and
you have entirely zinc.
00:26:08.250 --> 00:26:11.890
And so this is how you
make brass as an alloy.
00:26:11.890 --> 00:26:16.580
And what my other axis
is is the temperature.
00:26:16.580 --> 00:26:21.690
What you find is that there
is some kind of phase diagram
00:26:21.690 --> 00:26:29.920
such that you get a nice
mixture of copper and zinc
00:26:29.920 --> 00:26:32.180
only if you are at
high temperatures,
00:26:32.180 --> 00:26:34.070
whereas if you are
at low temperature,
00:26:34.070 --> 00:26:37.660
you basically will
separate into chunks
00:26:37.660 --> 00:26:40.950
that are rich copper and
chunks that are rich in zinc.
00:26:40.950 --> 00:26:44.710
And you'll have a
critical demixing
00:26:44.710 --> 00:26:50.560
point, which has exactly the
same properties as the Ising
00:26:50.560 --> 00:26:51.500
model.
00:26:51.500 --> 00:26:54.760
For example, this curve
will be characterized
00:26:54.760 --> 00:26:56.610
with an exponent
beta, which would
00:26:56.610 --> 00:26:59.811
be the beta of the Ising mode.
00:26:59.811 --> 00:27:05.410
And in particular, if I
were to take someplace
00:27:05.410 --> 00:27:10.080
in the vicinity of this and
try to write down a probability
00:27:10.080 --> 00:27:13.200
distribution, that
probability distribution
00:27:13.200 --> 00:27:15.960
would be exactly what
I have over there where
00:27:15.960 --> 00:27:22.470
m is, let's say, the difference
between the two types of alloys
00:27:22.470 --> 00:27:28.420
that I have compared to
each other over here.
00:27:28.420 --> 00:27:32.060
So this is related to 2x minus
1 or something like that.
00:27:34.880 --> 00:27:40.280
So as you go across your piece
of material close to here,
00:27:40.280 --> 00:27:42.190
there will be
compositional variations
00:27:42.190 --> 00:27:45.540
that are described by that.
00:27:45.540 --> 00:27:51.620
So the question is, I know
exactly what the probability
00:27:51.620 --> 00:27:56.120
distribution is for this system
to be an equilibrium given
00:27:56.120 --> 00:27:58.910
this choice of m.
00:27:58.910 --> 00:28:03.020
Again, with some set of
parameters, R, U, et cetera.
00:28:03.020 --> 00:28:07.730
The question is-- is the
dynamics again described
00:28:07.730 --> 00:28:10.350
by the same equation?
00:28:10.350 --> 00:28:12.810
And the answer is no.
00:28:12.810 --> 00:28:15.640
The same probability
of distribution
00:28:15.640 --> 00:28:19.430
can describe-- or
can be obtained
00:28:19.430 --> 00:28:21.250
with very different dynamics.
00:28:21.250 --> 00:28:23.640
And in particular,
what is happening
00:28:23.640 --> 00:28:28.970
in the system is that, if
I integrate this quantity m
00:28:28.970 --> 00:28:34.920
across the system, I will
get the total number of 1
00:28:34.920 --> 00:28:38.750
minus the other, which
is what is given to you,
00:28:38.750 --> 00:28:41.530
and it does not change
as a function of time.
00:28:41.530 --> 00:28:50.585
d by dt of this quantity is 0.
00:28:50.585 --> 00:28:53.058
It cannot change.
00:28:53.058 --> 00:28:53.970
OK.
00:28:53.970 --> 00:28:59.300
Whereas the equation that
I have written over here,
00:28:59.300 --> 00:29:03.630
in principle, locally, I
can, by adding the noise
00:29:03.630 --> 00:29:05.740
or by bringing things
from the neighborhood,
00:29:05.740 --> 00:29:08.135
I can change the value of m.
00:29:08.135 --> 00:29:09.320
I cannot do that.
00:29:09.320 --> 00:29:15.000
So this process of the
relaxation that would go
00:29:15.000 --> 00:29:19.100
on in data graphs cannot be
described by the time dependent
00:29:19.100 --> 00:29:22.270
on the Landau-Ginzburg
equation because you have this
00:29:22.270 --> 00:29:23.652
conservation here.
00:29:28.080 --> 00:29:30.260
OK.
00:29:30.260 --> 00:29:34.050
So what should we do?
00:29:34.050 --> 00:29:39.270
Well, when things are
conserved, like, say,
00:29:39.270 --> 00:29:42.580
as the gas particles
move in this fluid,
00:29:42.580 --> 00:29:45.460
and if I'm interested in
the number of particles
00:29:45.460 --> 00:29:48.390
in some cube, then the
change in the number
00:29:48.390 --> 00:29:51.050
of particles in some
cube in this room
00:29:51.050 --> 00:29:54.310
is related to the gradient
of the current that
00:29:54.310 --> 00:29:56.260
goes into that place.
00:29:56.260 --> 00:30:01.380
So the appropriate way of
writing an equation that
00:30:01.380 --> 00:30:04.710
describes, let's say, the
magnetization changing
00:30:04.710 --> 00:30:07.150
as a function of time.
00:30:07.150 --> 00:30:09.410
Given that you have
a conservation,
00:30:09.410 --> 00:30:14.340
though, is to write it
as minus the gradient
00:30:14.340 --> 00:30:15.670
of some kind of a current.
00:30:19.895 --> 00:30:22.980
So this j is some
kind of a current,
00:30:22.980 --> 00:30:25.045
and these would be vectors.
00:30:29.770 --> 00:30:32.720
This is a current
of the particles
00:30:32.720 --> 00:30:36.760
moving into the system.
00:30:36.760 --> 00:30:42.150
Now, in systems that
are dissipative,
00:30:42.150 --> 00:30:46.260
currents are related to the
gradient of some density
00:30:46.260 --> 00:30:49.210
through the diffusion
constant, et cetera.
00:30:49.210 --> 00:30:56.360
So it kind of makes sense to
imagine that this current is
00:30:56.360 --> 00:31:03.840
the gradient of something that
is trying-- or more precisely,
00:31:03.840 --> 00:31:06.780
minus the gradient
of something that
00:31:06.780 --> 00:31:10.790
tries to bring the system
to be, more or less,
00:31:10.790 --> 00:31:13.330
in its equilibrium state.
00:31:13.330 --> 00:31:18.740
Equilibrium state, as we said,
is determined by this data H,
00:31:18.740 --> 00:31:21.890
and we want to push
it in that direction.
00:31:21.890 --> 00:31:29.250
So we put our data
H by dm over here,
00:31:29.250 --> 00:31:32.283
and we put some kind
of a U over here.
00:31:35.670 --> 00:31:39.370
Of course, I would
have to add some kind
00:31:39.370 --> 00:31:43.180
of a conserved
random current also,
00:31:43.180 --> 00:31:47.640
which is the analog of
this non-conserved noise
00:31:47.640 --> 00:31:50.507
that I add over initially.
00:31:50.507 --> 00:31:51.007
OK.
00:31:55.350 --> 00:32:00.450
Now, the conservative version
of the equation, you can see,
00:32:00.450 --> 00:32:04.870
you have two more
derivatives with respect
00:32:04.870 --> 00:32:07.340
to what we had before.
00:32:07.340 --> 00:32:10.590
And so once I-- OK.
00:32:10.590 --> 00:32:19.530
So if we do something like
this, dm by dt is mu C.
00:32:19.530 --> 00:32:22.000
And then I would
have [INAUDIBLE]
00:32:22.000 --> 00:32:30.170
plus of R, rather
than by itself.
00:32:30.170 --> 00:32:35.880
Actually, it would be the plus
then of something like R m
00:32:35.880 --> 00:32:41.701
plus 4 U m cubed, and so forth.
00:32:41.701 --> 00:32:47.520
And we are going to
ignore this kind of term.
00:32:47.520 --> 00:32:50.150
And then there would
be high order terms
00:32:50.150 --> 00:32:53.735
that would show up minus
k the fourth derivative,
00:32:53.735 --> 00:32:55.130
and so forth.
00:32:59.320 --> 00:33:04.380
And then there may be some
kind of a conserved noise
00:33:04.380 --> 00:33:06.220
that I have to put outside.
00:33:11.055 --> 00:33:11.555
OK.
00:33:14.960 --> 00:33:18.890
So when I fully transform
this equation, what do I get?
00:33:18.890 --> 00:33:27.530
I will get that dm by dt is--
let's say in the full space,
00:33:27.530 --> 00:33:34.520
until there is a
function of qnt is minus
00:33:34.520 --> 00:33:39.770
U C. Because of this plus, then
there's an additional factor
00:33:39.770 --> 00:33:42.040
of q squared.
00:33:42.040 --> 00:33:48.540
And then I have R
plus kq squared plus L
00:33:48.540 --> 00:33:51.740
cubed to the fourth, et cetera.
00:33:51.740 --> 00:33:54.960
And then I will have a
fully transformed version
00:33:54.960 --> 00:33:58.529
of this conserved noise.
00:33:58.529 --> 00:33:59.028
OK.
00:34:02.280 --> 00:34:08.060
You can see that the difference
between this equation
00:34:08.060 --> 00:34:12.480
and the previous equation is
that all of the relaxation
00:34:12.480 --> 00:34:18.639
times will have an additional
factor of q squared.
00:34:18.639 --> 00:34:25.635
And so eventually, this shortest
relaxation time actually
00:34:25.635 --> 00:34:29.790
will glow like the size
of the system squared.
00:34:29.790 --> 00:34:31.522
Whereas previously,
it was saturated
00:34:31.522 --> 00:34:34.620
at the correlation length.
00:34:34.620 --> 00:34:37.639
And because you will
have this conservation,
00:34:37.639 --> 00:34:42.280
though, you have to
rearrange a lot of particles
00:34:42.280 --> 00:34:44.580
keeping their numbers constant.
00:34:44.580 --> 00:34:49.015
You have a much harder time
of relaxing the system.
00:34:49.015 --> 00:34:53.389
All of the relaxation
times, as we see,
00:34:53.389 --> 00:34:57.426
grow correspondingly
and become higher.
00:34:57.426 --> 00:34:58.310
OK.
00:34:58.310 --> 00:35:04.940
So indeed, for this class, one
can show that z starts with 4,
00:35:04.940 --> 00:35:07.940
and then there
will be corrections
00:35:07.940 --> 00:35:11.110
that would modify that.
00:35:11.110 --> 00:35:16.298
So the-- yes?
00:35:16.298 --> 00:35:19.376
AUDIENCE: How do we
define or how do we
00:35:19.376 --> 00:35:22.322
do a realization of the
conserved noise, conserved
00:35:22.322 --> 00:35:25.396
current-- conserved
noise in the room?
00:35:25.396 --> 00:35:26.146
MEHRAN KARDAR: OK.
00:35:26.146 --> 00:35:29.700
AUDIENCE: So it has some
kind of like correlation--
00:35:29.700 --> 00:35:32.790
self-correlation
properties, I suppose,
00:35:32.790 --> 00:35:36.080
because, if current
flowing out of some region,
00:35:36.080 --> 00:35:39.430
doesn't it want to go in?
00:35:39.430 --> 00:35:41.600
MEHRAN KARDAR: If
I go back here,
00:35:41.600 --> 00:35:46.270
I have a good idea of what is
happening because all I need,
00:35:46.270 --> 00:35:50.930
in order to ensure conservation,
is that the m by dt
00:35:50.930 --> 00:35:53.053
is the gradient of something.
00:35:53.053 --> 00:35:53.920
AUDIENCE: OK.
00:35:53.920 --> 00:35:58.015
MEHRAN KARDAR: So I can put
whatever I want over here.
00:35:58.015 --> 00:36:00.619
AUDIENCE: So if it's a scalar
or an [INAUDIBLE] field?
00:36:00.619 --> 00:36:01.410
MEHRAN KARDAR: Yes.
00:36:01.410 --> 00:36:04.019
As long as it is sitting
under the gradient--
00:36:04.019 --> 00:36:04.560
AUDIENCE: OK.
00:36:04.560 --> 00:36:07.610
MEHRAN KARDAR: --it will be OK,
which means this quantity here
00:36:07.610 --> 00:36:12.730
that I'm calling a to
z has a gradient in it.
00:36:12.730 --> 00:36:15.640
And if you wait for
about five minutes,
00:36:15.640 --> 00:36:19.030
we'll show that, because
of that in full space,
00:36:19.030 --> 00:36:22.590
rather than having--
well, I'll describe
00:36:22.590 --> 00:36:25.060
the difference between
non-conserved and conserved
00:36:25.060 --> 00:36:26.770
noise in fully space.
00:36:26.770 --> 00:36:30.046
It's much easier.
00:36:30.046 --> 00:36:30.546
OK.
00:36:35.010 --> 00:36:40.750
So actually, as far as what I
have discussed so far, which
00:36:40.750 --> 00:36:45.340
is relaxation, I don't
really need the noise
00:36:45.340 --> 00:36:47.790
because I can forget the noise.
00:36:47.790 --> 00:36:49.360
And all I have
said-- and I forgot
00:36:49.360 --> 00:36:54.170
the n tilde-- is that I
have a linear equation that
00:36:54.170 --> 00:36:56.230
relaxes your variable to 0.
00:36:56.230 --> 00:36:59.175
I can immediately read
off for the correlation,
00:36:59.175 --> 00:37:05.920
then this-- a correlation times
what I need the noise for it
00:37:05.920 --> 00:37:11.410
so that, ultimately, I don't go
to the medium is the potential,
00:37:11.410 --> 00:37:15.050
but I go to this
pro-rated distribution.
00:37:15.050 --> 00:37:20.880
So let's see what we have to
do in order to achieve that.
00:37:20.880 --> 00:37:24.235
For simplicity, let's
take this equation.
00:37:28.380 --> 00:37:34.940
Although I can take the
corresponding one for that.
00:37:34.940 --> 00:37:39.650
And let's calculate-- because
of the presence of this noise,
00:37:39.650 --> 00:37:44.260
if I run the same system
at different times,
00:37:44.260 --> 00:37:46.930
I will have different
realizations of the noise
00:37:46.930 --> 00:37:49.930
than if I had run many
versions of the system because
00:37:49.930 --> 00:37:52.020
of the realizations of noise.
00:37:52.020 --> 00:37:55.315
It's quantity and tilde
would be different.
00:37:55.315 --> 00:37:59.540
It would satisfy some kind
of a pro-rated distribution.
00:37:59.540 --> 00:38:02.480
So what I want to do is
to calculate averages,
00:38:02.480 --> 00:38:05.280
such as the average of m tilde.
00:38:05.280 --> 00:38:11.920
Let's say, q1 at time T
with m tilde q2 at time t.
00:38:15.340 --> 00:38:21.870
And you can see already
from this equation
00:38:21.870 --> 00:38:26.250
that, if I forget the part
that comes from the noise,
00:38:26.250 --> 00:38:28.755
whatever initial
condition that I have
00:38:28.755 --> 00:38:31.440
will eventually decay to 0.
00:38:31.440 --> 00:38:33.640
So the thing that
agitates and gives
00:38:33.640 --> 00:38:38.680
some kind of a randomness to
this really comes from this.
00:38:38.680 --> 00:38:40.620
So let's imagine
that we have looked
00:38:40.620 --> 00:38:43.620
at times that are
sufficiently long so
00:38:43.620 --> 00:38:45.880
that the influence of
the initial condition
00:38:45.880 --> 00:38:46.950
has died down.
00:38:46.950 --> 00:38:48.720
I don't want to
write the other term.
00:38:48.720 --> 00:38:53.570
I could do it, but it's kind
of boring to include it.
00:38:53.570 --> 00:38:57.340
So let's forget that and
focus on the integral.
00:38:57.340 --> 00:39:00.140
0 to t.
00:39:00.140 --> 00:39:04.230
Now, if I multiply two
of these quantities,
00:39:04.230 --> 00:39:08.318
I will have two
integrals over t prime.
00:39:08.318 --> 00:39:10.290
All right.
00:39:10.290 --> 00:39:16.625
Each one of them would decay
with the corresponding tau
00:39:16.625 --> 00:39:18.170
of q.
00:39:18.170 --> 00:39:20.260
In one case, tau of q1.
00:39:20.260 --> 00:39:27.080
In the other case, tau of q2.
00:39:27.080 --> 00:39:29.430
Coming from these things.
00:39:29.430 --> 00:39:35.780
And the noise, q1 at time
to 1 prime, and noise q2
00:39:35.780 --> 00:39:39.764
at time [? t2 prime. ?]
00:39:39.764 --> 00:39:41.760
OK.
00:39:41.760 --> 00:39:48.050
Now if I average over
the noise, then I
00:39:48.050 --> 00:39:51.414
have to do an average over here.
00:39:51.414 --> 00:39:51.913
OK.
00:39:55.140 --> 00:40:01.310
Now, one thing that I forgot to
mention right at the beginning
00:40:01.310 --> 00:40:04.146
is that, of course,
the average of this
00:40:04.146 --> 00:40:05.880
we are going to set to 0.
00:40:05.880 --> 00:40:08.020
It's the very least
that is important.
00:40:08.020 --> 00:40:09.520
Right?
00:40:09.520 --> 00:40:11.550
So if I do that,
clearly, the average
00:40:11.550 --> 00:40:14.280
of one of these in
full space would
00:40:14.280 --> 00:40:21.950
be 0 also because the full q is
related to the real space delta
00:40:21.950 --> 00:40:24.750
just by an integral.
00:40:24.750 --> 00:40:26.375
So if the average of
the integral is 0,
00:40:26.375 --> 00:40:29.390
the average of this is 0.
00:40:29.390 --> 00:40:33.200
So it turns out that, when
you look at the average of two
00:40:33.200 --> 00:40:36.010
of them-- and it's a
very simple exercise
00:40:36.010 --> 00:40:43.115
to just rewrite these things
in terms of 8 of x and t.
00:40:43.115 --> 00:40:47.490
8 of x and t applied
average that you have.
00:40:47.490 --> 00:40:52.840
And we find that the things that
are uncorrelated in real space
00:40:52.840 --> 00:40:56.280
also are uncoordinated
in full space.
00:40:56.280 --> 00:40:58.970
And so the variance
of this quantity
00:40:58.970 --> 00:41:05.000
is 2d delta 1 prime
minus [? d2 prime. ?]
00:41:05.000 --> 00:41:09.950
And then you have the analog
of the function in full space,
00:41:09.950 --> 00:41:16.330
which always carries the
solution of a factor of 2 pi
00:41:16.330 --> 00:41:17.370
to the d.
00:41:17.370 --> 00:41:20.000
And it becomes a sum
of the q's, as we've
00:41:20.000 --> 00:41:22.860
seen many times before.
00:41:22.860 --> 00:41:25.060
OK.
00:41:25.060 --> 00:41:28.350
So because of this
delta function,
00:41:28.350 --> 00:41:30.920
this delta integral
becomes one integral.
00:41:30.920 --> 00:41:34.480
So I have the integral 0 to t.
00:41:34.480 --> 00:41:39.170
The 2t prime I can
write as just 1t prime,
00:41:39.170 --> 00:41:42.990
and these two factors
merge into one.
00:41:42.990 --> 00:41:49.680
It comes into minus t minus
t prime over tau of q,
00:41:49.680 --> 00:41:53.650
except that I get
multiplied by a factor 2
00:41:53.650 --> 00:41:56.900
since I have two of them.
00:41:56.900 --> 00:42:05.640
And outside of the integral,
I will have this factor of 2d,
00:42:05.640 --> 00:42:10.875
and then 2 pi to the d delta
function with 1 plus 2.
00:42:15.125 --> 00:42:15.625
OK.
00:42:19.430 --> 00:42:21.880
Now, we are really
interested-- and I already
00:42:21.880 --> 00:42:24.825
kind of hinted at
that in the limit
00:42:24.825 --> 00:42:29.130
where time becomes very large.
00:42:29.130 --> 00:42:32.930
In the limit, where time
becomes a very large,
00:42:32.930 --> 00:42:36.630
essentially, I need to calculate
the limit of this integral
00:42:36.630 --> 00:42:38.308
as time becomes very large.
00:42:42.132 --> 00:42:44.680
And as time grows
to very large, this
00:42:44.680 --> 00:42:47.470
is just the integral
from 0 to infinity.
00:42:47.470 --> 00:42:55.280
And the integral is going
to give me 2 over tau of q.
00:42:55.280 --> 00:42:59.255
Essentially, you can see that
integrating at the upper end
00:42:59.255 --> 00:43:02.810
will give me just 0.
00:43:02.810 --> 00:43:05.445
Integrating at the
smaller end, it
00:43:05.445 --> 00:43:07.120
will be exponentially
small as it
00:43:07.120 --> 00:43:11.855
goes to infinity as a
factor of 2 over tau.
00:43:11.855 --> 00:43:12.750
OK.
00:43:12.750 --> 00:43:14.202
Yes?
00:43:14.202 --> 00:43:18.460
AUDIENCE: Are you
assuming or-- yeah.
00:43:18.460 --> 00:43:22.230
Are you assuming that
tau of q is even in q
00:43:22.230 --> 00:43:24.039
to be able to combine to--
00:43:24.039 --> 00:43:24.830
MEHRAN KARDAR: Yes.
00:43:24.830 --> 00:43:25.320
AUDIENCE: --2 tau?
00:43:25.320 --> 00:43:25.850
So-- OK.
00:43:25.850 --> 00:43:28.183
MEHRAN KARDAR: I'm thinking
of the tau of q's that we've
00:43:28.183 --> 00:43:29.201
calculated over here.
00:43:29.201 --> 00:43:29.742
AUDIENCE: OK.
00:43:29.742 --> 00:43:30.990
MEHRAN KARDAR: OK.
00:43:30.990 --> 00:43:31.820
Yes?
00:43:31.820 --> 00:43:34.850
AUDIENCE: The e
times the [INAUDIBLE]
00:43:34.850 --> 00:43:36.370
do the equation of something?
00:43:36.370 --> 00:43:37.980
Right?
00:43:37.980 --> 00:43:42.030
MEHRAN KARDAR: At this stage, I
am focusing on this expression
00:43:42.030 --> 00:43:44.610
over here, where a is not.
00:43:44.610 --> 00:43:46.640
But I will come to
that expression also.
00:43:49.900 --> 00:43:54.440
So for the time being, this
d is just the same constant
00:43:54.440 --> 00:43:56.455
as we have over here.
00:43:56.455 --> 00:43:58.200
OK?
00:43:58.200 --> 00:44:01.310
So you can see that
the final answer
00:44:01.310 --> 00:44:08.680
is going to be d
over tau of q 2 pi
00:44:08.680 --> 00:44:12.314
to the d delta
function q1 plus q2.
00:44:15.150 --> 00:44:21.700
And if I use the value of tau
of q that I have, tau of q
00:44:21.700 --> 00:44:29.820
is mu R plus kq squared,
which becomes d over mu R
00:44:29.820 --> 00:44:33.600
plus kq squared and so forth.
00:44:33.600 --> 00:44:38.310
2 pi to the d delta
function q1 plus q2.
00:44:42.340 --> 00:44:49.510
So essentially, if I take this
linearized time dependent line
00:44:49.510 --> 00:44:54.180
of Landau-Ginzburg equation,
run it for a very long time,
00:44:54.180 --> 00:44:56.970
and look at the
correlations of the field,
00:44:56.970 --> 00:44:59.340
I see that the
correlations of the field,
00:44:59.340 --> 00:45:03.380
at the limit of long times,
satisfy this expression.
00:45:06.220 --> 00:45:10.090
Now, what do I know if
I look at the top line
00:45:10.090 --> 00:45:14.380
that I have for the
probability of distribution?
00:45:14.380 --> 00:45:17.890
I can go and express that
probability of distribution
00:45:17.890 --> 00:45:19.175
in full mode.
00:45:19.175 --> 00:45:22.160
In the linear
version, I immediately
00:45:22.160 --> 00:45:27.580
get that the probability
of m tilde of q
00:45:27.580 --> 00:45:31.960
is proportional to a
product over different q's,
00:45:31.960 --> 00:45:36.890
e to the minus R
plus kq squared,
00:45:36.890 --> 00:45:42.250
et cetera, and tilde
of q squared over 2.
00:45:42.250 --> 00:45:44.600
All right.
00:45:44.600 --> 00:45:48.060
So when I look at the
equilibrium linear
00:45:48.060 --> 00:45:51.730
as Landau-Ginzburg, I can
see that, if I calculate
00:45:51.730 --> 00:45:57.560
the average of m of
q1, m of q2, then
00:45:57.560 --> 00:46:01.950
this is an equilibrium average.
00:46:01.950 --> 00:46:06.980
What I would get is 2 pi
to the d delta function
00:46:06.980 --> 00:46:10.870
of q1 plus q2 because, clearly,
the different q's are recovered
00:46:10.870 --> 00:46:12.620
from each other.
00:46:12.620 --> 00:46:17.530
And for the particular value of
q, what I will get is 1 over R
00:46:17.530 --> 00:46:21.030
plus kq squared and so forth.
00:46:21.030 --> 00:46:21.890
Yes?
00:46:21.890 --> 00:46:26.235
AUDIENCE: So isn't the 2
over tau-- [INAUDIBLE]?
00:46:26.235 --> 00:46:27.026
MEHRAN KARDAR: Yes.
00:46:36.132 --> 00:46:37.124
Over 2.
00:46:43.072 --> 00:46:43.572
Yeah.
00:46:54.976 --> 00:46:55.476
OK.
00:47:02.420 --> 00:47:03.412
OK.
00:47:03.412 --> 00:47:07.975
I changed because I
had 2d cancels the 2.
00:47:07.975 --> 00:47:10.600
I would have to put
here tau of q over 2.
00:47:10.600 --> 00:47:13.360
I had 2d times tau of q over 2.
00:47:13.360 --> 00:47:15.100
So it's d tau.
00:47:15.100 --> 00:47:18.618
And inverse of tau, I
have everything correct.
00:47:18.618 --> 00:47:21.330
OK.
00:47:21.330 --> 00:47:28.031
So if you make an even number
of errors, the answer comes up.
00:47:28.031 --> 00:47:28.530
OK.
00:47:28.530 --> 00:47:31.700
But you can now
compare this expression
00:47:31.700 --> 00:47:36.940
that comes from equilibrium
and this expression that
00:47:36.940 --> 00:47:42.760
comes from the long time
limit of this noisy equation.
00:47:42.760 --> 00:47:48.120
OK So we want to
choose our noise
00:47:48.120 --> 00:47:55.590
so that the stochastic
dynamics gives the same value
00:47:55.590 --> 00:48:01.100
as equilibrium, just like we
did for the case of a Brownian
00:48:01.100 --> 00:48:06.340
particular where you have some
kind of an Einstein equation
00:48:06.340 --> 00:48:15.125
that was relating the strength
of the noise and the mobility.
00:48:15.125 --> 00:48:19.120
And we see that here
all I need to do
00:48:19.120 --> 00:48:27.198
is to ensure that d over
mu should be equal to 1.
00:48:27.198 --> 00:48:27.698
OK.
00:48:33.011 --> 00:48:41.810
Now, the thing is that,
if I am doing this,
00:48:41.810 --> 00:48:48.430
I, in principle, can have a
different noise for each q,
00:48:48.430 --> 00:48:52.110
and compensate by different
mobility for each q.
00:48:52.110 --> 00:48:55.350
And I would get the same answer.
00:48:55.350 --> 00:49:01.690
So in the non-conserved version
of this time dependent dynamics
00:49:01.690 --> 00:49:07.300
that you wrote down,
the d was a constant
00:49:07.300 --> 00:49:11.280
and the mu was a constant.
00:49:11.280 --> 00:49:14.940
Whereas, if you want to get
the same equilibrium result out
00:49:14.940 --> 00:49:19.300
of the conserved dynamics,
you can see that, essentially,
00:49:19.300 --> 00:49:23.650
what we previously had as
mu became something that
00:49:23.650 --> 00:49:26.540
is proportional to q squared.
00:49:26.540 --> 00:49:31.540
So essentially, here, this
becomes mu C q squared.
00:49:31.540 --> 00:49:35.070
So clearly, in order
to get the same answer,
00:49:35.070 --> 00:49:42.750
I have to put my noise to be
proportional to q squared also.
00:49:42.750 --> 00:49:46.570
And we can see that this kind of
conserved noise that I put over
00:49:46.570 --> 00:49:49.990
here achieves that
because, as I said,
00:49:49.990 --> 00:49:54.870
this conserved noise is the
gradient of something, which
00:49:54.870 --> 00:49:58.090
means that, when I go to
fully space, if it be q,
00:49:58.090 --> 00:50:00.740
it will be proportional to q.
00:50:00.740 --> 00:50:02.620
And when I take its
variants, it's variants
00:50:02.620 --> 00:50:05.800
will be proportional
to q squared.
00:50:05.800 --> 00:50:09.580
Anything precisely canceled.
00:50:09.580 --> 00:50:12.580
But you can see
that you also-- this
00:50:12.580 --> 00:50:16.390
had a physical explanation in
terms of a conservation law.
00:50:16.390 --> 00:50:20.450
In principle, you can cook
up all kinds of b of q and mu
00:50:20.450 --> 00:50:22.070
of q.
00:50:22.070 --> 00:50:26.780
As long as this equality is
satisfied, you will have,
00:50:26.780 --> 00:50:31.010
for these linear
stochastic equations,
00:50:31.010 --> 00:50:35.381
the guarantee that you would
always get the same equilibrium
00:50:35.381 --> 00:50:35.880
result.
00:50:35.880 --> 00:50:38.700
Because if you wait
for this dynamics
00:50:38.700 --> 00:50:43.060
to settle down after long times,
you will get to the answer.
00:50:43.060 --> 00:50:43.842
Yes?
00:50:43.842 --> 00:50:48.312
AUDIENCE: I wonder how general
is this result for stochastic
00:50:48.312 --> 00:50:50.070
[INAUDIBLE]?
00:50:50.070 --> 00:50:50.820
MEHRAN KARDAR: OK.
00:50:50.820 --> 00:50:51.736
AUDIENCE: But what I--
00:50:51.736 --> 00:50:54.380
MEHRAN KARDAR: So
what I showed you
00:50:54.380 --> 00:50:58.570
was with for linearized version,
and the only thing that I
00:50:58.570 --> 00:51:01.220
calculated was the variance.
00:51:01.220 --> 00:51:04.090
And I showed that the
variances were the same.
00:51:04.090 --> 00:51:06.310
And if I have a Gaussian
problem of distribution,
00:51:06.310 --> 00:51:08.920
the variance is
completely categorizable
00:51:08.920 --> 00:51:10.360
with distribution.
00:51:10.360 --> 00:51:11.880
So this is safe.
00:51:11.880 --> 00:51:15.580
But we are truly interested
in the more general
00:51:15.580 --> 00:51:18.740
non-Gaussian probability
of distribution.
00:51:18.740 --> 00:51:24.440
So the question really is-- if
I keep the full non-linearity
00:51:24.440 --> 00:51:29.730
in this story, would
I be able to show
00:51:29.730 --> 00:51:31.830
that the probability
of distribution that
00:51:31.830 --> 00:51:36.140
will be characterized
by all kinds of moments
00:51:36.140 --> 00:51:38.472
eventually has the
same behavior as that.
00:51:38.472 --> 00:51:39.180
AUDIENCE: Mm-hmm.
00:51:39.180 --> 00:51:41.460
MEHRAN KARDAR: And the
answer is, in fact, yes.
00:51:41.460 --> 00:51:46.310
There's a procedure that
relies on converting
00:51:46.310 --> 00:51:48.940
this equation-- sorry.
00:51:48.940 --> 00:51:51.660
One equation that
governs the evolution
00:51:51.660 --> 00:51:55.421
of the full probability
as a function of time.
00:51:55.421 --> 00:51:55.920
Right?
00:51:55.920 --> 00:52:00.930
So basically, I can start
with a an initial probability
00:52:00.930 --> 00:52:04.430
and see how this probability
evolves as a function of time.
00:52:04.430 --> 00:52:08.220
And this is sometimes
called a master equation.
00:52:08.220 --> 00:52:10.255
Sometimes, called a
[INAUDIBLE] equation.
00:52:13.000 --> 00:52:20.790
And we did cover, in fact,
this in-- next spring
00:52:20.790 --> 00:52:23.370
in the statistical
physics and biology,
00:52:23.370 --> 00:52:25.630
we spent some time talking
about these things.
00:52:25.630 --> 00:52:29.390
So you can come back to the
third version of this class.
00:52:29.390 --> 00:52:35.015
And one can ensure that,
with appropriate choice
00:52:35.015 --> 00:52:40.270
of the noise, the asymptotic
solution for this probability
00:52:40.270 --> 00:52:43.880
distribution is
whatever Landau-Ginzburg
00:52:43.880 --> 00:52:45.600
or other probability
distribution
00:52:45.600 --> 00:52:47.458
that you need most.
00:52:47.458 --> 00:52:51.130
AUDIENCE: So is this
true if we assume
00:52:51.130 --> 00:52:54.140
Landau-Ginzburg potential
for how resistant?
00:52:54.140 --> 00:52:54.975
MEHRAN KARDAR: Yes.
00:52:54.975 --> 00:52:55.965
AUDIENCE: OK.
00:52:55.965 --> 00:52:59.200
Maybe this is not a very
good-stated question,
00:52:59.200 --> 00:53:01.705
but is there kind of like
an even more general level?
00:53:01.705 --> 00:53:03.330
MEHRAN KARDAR: I'll
come to that, sure.
00:53:03.330 --> 00:53:06.390
But currently, the way
that I set up the problem
00:53:06.390 --> 00:53:09.110
was that we know some
complicated equilibrium
00:53:09.110 --> 00:53:13.250
force that exist-- form of
the probability that exist.
00:53:13.250 --> 00:53:16.510
And these kinds of
linear-- these kinds
00:53:16.510 --> 00:53:20.090
of stochastic linear
or non-linear evolution
00:53:20.090 --> 00:53:22.340
equations-- generally
called non- [INAUDIBLE]
00:53:22.340 --> 00:53:26.150
equations-- one can show that,
with the appropriate choice
00:53:26.150 --> 00:53:30.690
of the noise, we'll be able
to asymptotically reproduce
00:53:30.690 --> 00:53:34.192
the probability of
distribution that we knew.
00:53:34.192 --> 00:53:35.894
But now, the question
is, of course, you
00:53:35.894 --> 00:53:37.685
don't know the probability
of distribution.
00:53:37.685 --> 00:53:40.113
And I'll say a few
words about that.
00:53:40.113 --> 00:53:40.654
AUDIENCE: OK.
00:53:40.654 --> 00:53:41.570
Thank you.
00:53:41.570 --> 00:53:44.540
MEHRAN KARDAR: Anything else?
00:53:44.540 --> 00:53:45.420
OK.
00:53:45.420 --> 00:53:52.130
So the lesson of this
part is that the field
00:53:52.130 --> 00:53:58.070
of dynamic or critical phenomena
is quite rich, much richer
00:53:58.070 --> 00:54:01.450
than the corresponding
equilibrium critical phenomena
00:54:01.450 --> 00:54:04.610
because the same
equilibrium state can
00:54:04.610 --> 00:54:08.360
be obtained by various
different types of dynamics.
00:54:08.360 --> 00:54:12.400
And I explained to you
just one conservation law,
00:54:12.400 --> 00:54:14.090
but there could be
some combination
00:54:14.090 --> 00:54:17.020
of conservation of energy,
conservation of something.
00:54:17.020 --> 00:54:21.360
So there is a whole listing of
different universality classes
00:54:21.360 --> 00:54:26.254
that people have targeted
for the dynamics.
00:54:26.254 --> 00:54:30.620
But not all of this
was assuming that you
00:54:30.620 --> 00:54:35.800
know what the ultimate answer
is because, in all cases,
00:54:35.800 --> 00:54:38.870
the equations that
we're writing are
00:54:38.870 --> 00:54:43.910
dependent on some kind of a
gradient descent conserved
00:54:43.910 --> 00:54:47.100
or non-conserved around
something that corresponded
00:54:47.100 --> 00:54:49.460
to the log of probability
of distribution
00:54:49.460 --> 00:54:52.190
that we eventually want to put.
00:54:52.190 --> 00:54:55.770
And maybe you don't
know that, and so
00:54:55.770 --> 00:55:04.000
let me give you a particular
example in the context of,
00:55:04.000 --> 00:55:06.464
let's say, surface
interface fluctuations.
00:55:12.160 --> 00:55:15.160
Starting from things that
you know and then building
00:55:15.160 --> 00:55:18.780
to something that
maybe you don't.
00:55:18.780 --> 00:55:21.888
Let's first with the
case of a soap bubble.
00:55:25.720 --> 00:55:30.680
So we take some kind of
a circle or whatever,
00:55:30.680 --> 00:55:34.980
and we put a soap
bubble on top of it.
00:55:34.980 --> 00:55:41.080
And in this case, the
energy of the formation--
00:55:41.080 --> 00:55:43.780
the cost of the formation
comes from surface tension.
00:55:50.020 --> 00:55:54.900
And let's say, the
cost of the deformation
00:55:54.900 --> 00:55:58.830
is the changing area
times some sigma.
00:56:04.450 --> 00:56:10.160
So I neglect the
contribution that
00:56:10.160 --> 00:56:13.870
comes from the flat surface,
and see if I make a deformation.
00:56:13.870 --> 00:56:17.460
If I make a deformation, I have
changed the area of the spin.
00:56:17.460 --> 00:56:20.220
So there is a cost that is
proportion to the surface
00:56:20.220 --> 00:56:22.930
tension times the
change in area.
00:56:22.930 --> 00:56:26.230
Change in area locally
is the square root of 1
00:56:26.230 --> 00:56:29.240
plus the gradient
of a height profile.
00:56:29.240 --> 00:56:34.750
So what I can do is I can define
at each point on the surface
00:56:34.750 --> 00:56:39.420
how much it has changed its
height from being perfectly
00:56:39.420 --> 00:56:39.920
flat.
00:56:39.920 --> 00:56:42.320
So h equals to 0 is flat.
00:56:42.320 --> 00:56:47.770
Local area is the integral
dx dy of square root of 1
00:56:47.770 --> 00:56:50.790
plus gradient of h
squared, minus 1.
00:56:50.790 --> 00:56:53.060
That corresponds to the flat.
00:56:53.060 --> 00:56:54.420
And so then you expand that.
00:56:54.420 --> 00:56:58.360
The first term is going to
be the integral gradient of h
00:56:58.360 --> 00:56:58.860
squared.
00:57:01.500 --> 00:57:06.490
So this is the analog of
what we had over there, only
00:57:06.490 --> 00:57:07.640
the first term.
00:57:07.640 --> 00:57:10.440
So you would say that the
equation that you would write
00:57:10.440 --> 00:57:16.210
down for this would be
all to some constant mu
00:57:16.210 --> 00:57:18.690
proportional to the
variations of this,
00:57:18.690 --> 00:57:24.260
which will give me something
like sigma Laplacian of h.
00:57:24.260 --> 00:57:27.970
But because of the particles
from the air constantly
00:57:27.970 --> 00:57:30.380
bombarding the
surface, there will
00:57:30.380 --> 00:57:32.897
be some noise that
depends on where
00:57:32.897 --> 00:57:36.108
you are on the surface in time.
00:57:36.108 --> 00:57:40.335
And this is the
non-conserved version.
00:57:44.810 --> 00:57:48.030
And you can from
this very quickly get
00:57:48.030 --> 00:57:53.610
that the expectation value
of h tilde of q squared
00:57:53.610 --> 00:57:58.730
is going to be proportional
to something like D
00:57:58.730 --> 00:58:03.840
over mu sigma q squared,
because of this q squared.
00:58:03.840 --> 00:58:06.990
And if you ask how
much fluctuations you
00:58:06.990 --> 00:58:10.170
have in real space--
so that typical scale
00:58:10.170 --> 00:58:13.620
of the fluctuations
in real space--
00:58:13.620 --> 00:58:18.010
will come from integrating
1 over q squared.
00:58:18.010 --> 00:58:21.110
And it's going to be
our usual things that
00:58:21.110 --> 00:58:23.330
have this logarithmic
dependence, so there will
00:58:23.330 --> 00:58:26.850
be something that ultimately
will go logarithmically
00:58:26.850 --> 00:58:28.960
with the size of the system.
00:58:28.960 --> 00:58:31.440
The constant of
proportionality will be
00:58:31.440 --> 00:58:34.080
proportional to kt over sigma.
00:58:34.080 --> 00:58:38.900
So you have to choose your D
and mu to correspond to this.
00:58:38.900 --> 00:58:42.750
But basically, a soap
film, as an example
00:58:42.750 --> 00:58:45.970
of all kinds of Goldstone
mode-like things
00:58:45.970 --> 00:58:47.410
that we have seen.
00:58:47.410 --> 00:58:49.890
It's a 2-dimensional entity.
00:58:49.890 --> 00:58:53.120
We will have logarithmic
fluctuations-- not very big,
00:58:53.120 --> 00:58:56.560
but ultimately, at
large enough distances,
00:58:56.560 --> 00:58:59.110
it will have fluctuations.
00:58:59.110 --> 00:59:02.230
So that was non-conserved.
00:59:02.230 --> 00:59:04.750
I can imagine that,
rather than this,
00:59:04.750 --> 00:59:12.150
I have the case of
a surface of a pool.
00:59:12.150 --> 00:59:17.890
So here I have some
depth of water,
00:59:17.890 --> 00:59:21.930
and then there's the surface
of the pool of water.
00:59:21.930 --> 00:59:26.730
And the difference between this
case and the previous case--
00:59:26.730 --> 00:59:29.530
both of them can be described
by a height function.
00:59:33.130 --> 00:59:36.280
The difference is that
if I ignore evaporation
00:59:36.280 --> 00:59:41.020
and condensation, the
total mass of water
00:59:41.020 --> 00:59:42.800
is going to be conserved.
00:59:42.800 --> 00:59:49.420
So I would need to have divided
t of the integral dx d qx
00:59:49.420 --> 00:59:55.060
h of x and t to be 0.
00:59:55.060 --> 00:59:57.746
So this would go into
the conserved variety.
01:00:00.500 --> 01:00:05.120
And while, if I create a
ripple on the surface of this
01:00:05.120 --> 01:00:10.770
compared to the surface of
that, the relaxation time
01:00:10.770 --> 01:00:13.320
through this
dissipative dynamics
01:00:13.320 --> 01:00:17.280
would be much longer in this
case as opposed to that case.
01:00:17.280 --> 01:00:20.390
Ultimately, if I wait
sufficiently long time,
01:00:20.390 --> 01:00:24.640
both of them would have
exactly the same fluctuations.
01:00:24.640 --> 01:00:27.710
That is, you would go
logarithmically with the length
01:00:27.710 --> 01:00:31.050
scale over which [INAUDIBLE].
01:00:31.050 --> 01:00:37.170
OK, so now let's look at
another system that fluctuates.
01:00:37.170 --> 01:00:40.620
And I don't know what
the final answer is.
01:00:40.620 --> 01:00:43.240
That was the question,
maybe, that you asked.
01:00:43.240 --> 01:00:46.620
The example that I will
give is the following--
01:00:46.620 --> 01:00:50.480
so suppose that
you have a surface.
01:00:50.480 --> 01:00:56.330
And then you have a
rain of sticky materials
01:00:56.330 --> 01:00:59.670
that falls down on top of it.
01:00:59.670 --> 01:01:03.500
So this material will come down.
01:01:03.500 --> 01:01:05.400
You'll have something like this.
01:01:05.400 --> 01:01:07.860
And then as time
goes on, there will
01:01:07.860 --> 01:01:12.090
be more material that will come,
more material that will come,
01:01:12.090 --> 01:01:14.210
more material that will come.
01:01:14.210 --> 01:01:20.840
So there, because the particles
are raining down randomly
01:01:20.840 --> 01:01:25.810
at different points, there
will be a stochastic process
01:01:25.810 --> 01:01:27.450
that is going on.
01:01:27.450 --> 01:01:32.230
So you can try to
characterize the system
01:01:32.230 --> 01:01:35.360
in terms of a height that
changes as a function of t
01:01:35.360 --> 01:01:38.890
and as a function of position.
01:01:38.890 --> 01:01:44.930
And there could be all kinds
of microscopic things going on,
01:01:44.930 --> 01:01:46.930
like maybe these
are particles that
01:01:46.930 --> 01:01:51.260
are representing some kind
of a deposition process.
01:01:51.260 --> 01:01:54.390
And then they come, they
stick in a particular way.
01:01:54.390 --> 01:01:56.580
Maybe they can slide
on the surface.
01:01:56.580 --> 01:02:01.360
We can imagine all kinds of
microscopic degrees of freedom
01:02:01.360 --> 01:02:04.200
and things that we can put.
01:02:04.200 --> 01:02:08.660
But you say, well, can
I change my perspective,
01:02:08.660 --> 01:02:12.500
and try to describe the
system the same way that we
01:02:12.500 --> 01:02:16.180
did for the case of
coarse grading and going
01:02:16.180 --> 01:02:18.010
from without the
microscopic details
01:02:18.010 --> 01:02:22.930
to describe the phenomenological
Landau-Ginzburg equation?
01:02:22.930 --> 01:02:26.910
And so you say, OK, there
is a height that is growing.
01:02:26.910 --> 01:02:30.660
And what I will write
down is an equation
01:02:30.660 --> 01:02:32.510
that is very similar
to the equations
01:02:32.510 --> 01:02:36.090
that I had written before.
01:02:36.090 --> 01:02:40.340
Now I'm going to follow
the same kind of reasoning
01:02:40.340 --> 01:02:44.570
that we did in the construction
of this Landau-Ginzburg model,
01:02:44.570 --> 01:02:49.350
is we said that this weight
is going to depend on all
01:02:49.350 --> 01:02:52.660
kinds of things that
relate to this height
01:02:52.660 --> 01:02:53.870
that I don't quite know.
01:02:53.870 --> 01:02:58.230
So let's imagine that there
is some kind of a function
01:02:58.230 --> 01:03:01.570
of the height itself.
01:03:01.570 --> 01:03:04.420
And potentially, just
like we did over there,
01:03:04.420 --> 01:03:08.176
the gradient of the height,
five derivatives of the height,
01:03:08.176 --> 01:03:08.675
et cetera.
01:03:12.460 --> 01:03:16.280
And then I will start to
make an expansion of this
01:03:16.280 --> 01:03:20.750
in the same spirit that I did
for the Landau-Ginzburg Model,
01:03:20.750 --> 01:03:23.540
except that when I was doing
the Landau-Ginzburg Model,
01:03:23.540 --> 01:03:26.840
I was doing the expansion
at the level of looking
01:03:26.840 --> 01:03:30.450
at the probability distribution
and the log of the probability.
01:03:30.450 --> 01:03:34.625
Here I'm making the expansion
at the level of an equation that
01:03:34.625 --> 01:03:35.500
governs the dynamics.
01:03:38.580 --> 01:03:41.250
Of course, in this
particular system,
01:03:41.250 --> 01:03:44.620
that's not the end of story,
because the change in height
01:03:44.620 --> 01:03:49.600
is also governed by this random
addition of the particles.
01:03:49.600 --> 01:03:53.080
So there is some
function that changes
01:03:53.080 --> 01:03:55.070
as a function of
position and time,
01:03:55.070 --> 01:03:57.040
depending on whether,
at that time,
01:03:57.040 --> 01:03:59.370
a particle was dropped down.
01:03:59.370 --> 01:04:01.920
I can always take
the average of this
01:04:01.920 --> 01:04:07.740
to be 0, and put that average
into the expansion of this
01:04:07.740 --> 01:04:10.220
starting from a constant.
01:04:10.220 --> 01:04:13.200
Basically, if I just
have a single point
01:04:13.200 --> 01:04:17.010
and I randomly drop particles
at that single point,
01:04:17.010 --> 01:04:19.560
there will be an
average growth velocity,
01:04:19.560 --> 01:04:21.850
an average addition
to the height,
01:04:21.850 --> 01:04:23.530
that averages over here.
01:04:23.530 --> 01:04:27.580
But there will be fluctuations
that are going [INAUDIBLE].
01:04:27.580 --> 01:04:31.360
OK but the constant
is the first term
01:04:31.360 --> 01:04:34.330
in an expansion such as this.
01:04:34.330 --> 01:04:38.220
And you can start thinking,
OK, what next order?
01:04:38.220 --> 01:04:41.300
Can I put something
like alpha h?
01:04:41.300 --> 01:04:43.840
Potentially-- depends
on your system--
01:04:43.840 --> 01:04:49.440
but if the system is invariant
whether you started from here
01:04:49.440 --> 01:04:52.490
or whether you started from
there-- something like gravity,
01:04:52.490 --> 01:04:55.540
for example, is not
important-- you say, OK,
01:04:55.540 --> 01:05:00.110
I cannot have any function of
h if my dynamics will proceed
01:05:00.110 --> 01:05:05.330
exactly the same way if I
were to translate this surface
01:05:05.330 --> 01:05:07.590
to some further up
or further down.
01:05:07.590 --> 01:05:10.760
If I see that there's no change
in future dynamics on average,
01:05:10.760 --> 01:05:14.660
then the dynamic
cannot depend on this.
01:05:14.660 --> 01:05:16.970
OK, so we've got rid of that.
01:05:16.970 --> 01:05:19.370
And any function
of h-- can I put
01:05:19.370 --> 01:05:23.640
something that is
proportional to gradient of h?
01:05:23.640 --> 01:05:28.070
Maybe for something I can,
but for h itself I cannot,
01:05:28.070 --> 01:05:30.510
because h is a scalar gradient.
01:05:30.510 --> 01:05:33.990
If h is a vector, I
can't set something
01:05:33.990 --> 01:05:40.620
that is a scalar equal to a
vector, so I can't have this.
01:05:40.620 --> 01:05:41.914
Yes?
01:05:41.914 --> 01:05:43.455
AUDIENCE: Couldn't
you, in principle,
01:05:43.455 --> 01:05:47.130
make your constant term in
front of the gradient also
01:05:47.130 --> 01:05:48.570
a vector and [INAUDIBLE]?
01:05:50.659 --> 01:05:51.700
MEHRAN KARDAR: You could.
01:05:51.700 --> 01:05:54.690
So there's a whole set
of different systems
01:05:54.690 --> 01:05:56.630
that you can be thinking about.
01:05:56.630 --> 01:06:00.220
Right now, I want to focus
on the simplest system, which
01:06:00.220 --> 01:06:02.770
is a scalar field, so
that my equation can
01:06:02.770 --> 01:06:05.404
be as simple as
possible, but still we
01:06:05.404 --> 01:06:07.070
will see it has
sufficient complication.
01:06:10.640 --> 01:06:12.940
So you can see that
if I don't have them,
01:06:12.940 --> 01:06:15.770
the next order term
that I can have
01:06:15.770 --> 01:06:17.480
would be something
like a Laplacian.
01:06:20.560 --> 01:06:24.440
So this kind of diffusion
equation, you can see,
01:06:24.440 --> 01:06:28.780
has to emerge as a low-order
expansion of something
01:06:28.780 --> 01:06:29.280
like this.
01:06:29.280 --> 01:06:32.140
And this is the ubiquity of the
diffusion equation appearing
01:06:32.140 --> 01:06:34.380
all over the place.
01:06:34.380 --> 01:06:36.220
And then you could
have terms that
01:06:36.220 --> 01:06:38.890
would be of the order of
the fourth derivative,
01:06:38.890 --> 01:06:40.520
and so forth.
01:06:40.520 --> 01:06:42.480
There's nothing wrong with that.
01:06:46.410 --> 01:06:49.940
And then, if you
think about it, you'll
01:06:49.940 --> 01:06:55.420
see that there is one
interesting possibility that
01:06:55.420 --> 01:06:58.980
is not allowed for
that system, but is
01:06:58.980 --> 01:07:03.950
allowed for this system, which
is something that is a scalar.
01:07:03.950 --> 01:07:06.366
It's the gradient of h squared.
01:07:09.320 --> 01:07:15.100
Now I could not have added this
term for the case of the soap
01:07:15.100 --> 01:07:17.960
bubble for the
following reason--
01:07:17.960 --> 01:07:22.760
that if I reverse the
soap bubble so that h
01:07:22.760 --> 01:07:27.050
becomes minus h, the dynamics
would proceed exactly
01:07:27.050 --> 01:07:28.960
as before.
01:07:28.960 --> 01:07:34.020
So the soap bubble has a
symmetry of h going to minus h,
01:07:34.020 --> 01:07:38.160
and so that symmetry should
be preserved in the equation.
01:07:38.160 --> 01:07:41.820
This term breaks that symmetry
because the left-hand side
01:07:41.820 --> 01:07:45.760
is odd in h, whereas the
right-hand side of this term
01:07:45.760 --> 01:07:48.080
would be even in h.
01:07:48.080 --> 01:07:51.990
But for the case of the growing
surface-- and you've seen
01:07:51.990 --> 01:07:53.140
things that are growing.
01:07:53.140 --> 01:07:55.520
And typically, if I
give you something
01:07:55.520 --> 01:07:59.270
that has grown like the
tree trunk, for example,
01:07:59.270 --> 01:08:02.400
and if I take the
picture of a part of it,
01:08:02.400 --> 01:08:05.370
and you don't see where the
center is, where the end is,
01:08:05.370 --> 01:08:08.040
you can immediately
tell from the way
01:08:08.040 --> 01:08:11.760
that the shape of this
object is, that it is growing
01:08:11.760 --> 01:08:13.960
in some particular direction.
01:08:13.960 --> 01:08:17.590
So for growth systems, that
symmetry does not exist.
01:08:17.590 --> 01:08:22.220
You are allowed to have
this term, and so forth.
01:08:22.220 --> 01:08:26.300
Now the interesting
thing about this term
01:08:26.300 --> 01:08:31.465
is that there is no beta
h that you can write down
01:08:31.465 --> 01:08:36.040
that is local-- some
function of h such
01:08:36.040 --> 01:08:40.229
that if you take a functional
derivative with respect to h,
01:08:40.229 --> 01:08:46.310
it will reproduce that
term-- just does not exist.
01:08:46.310 --> 01:08:51.750
So you can see that
somehow immediately,
01:08:51.750 --> 01:08:59.380
as soon as we liberate ourselves
from writing equations that
01:08:59.380 --> 01:09:03.899
came from functional
derivative of something,
01:09:03.899 --> 01:09:07.310
but potentially have
physical significance,
01:09:07.310 --> 01:09:10.810
we can write down new terms.
01:09:10.810 --> 01:09:14.000
So this is actually--
also, you can do
01:09:14.000 --> 01:09:17.020
this, even for two particles.
01:09:17.020 --> 01:09:22.770
A potential v of x1 and x2 will
have some kind of derivatives.
01:09:22.770 --> 01:09:24.505
But if you write
dynamical equations,
01:09:24.505 --> 01:09:26.600
there are dynamical
equations that
01:09:26.600 --> 01:09:31.080
allow you to rotate
1 x from x1 to x2.
01:09:31.080 --> 01:09:32.759
That kind of term
will never come
01:09:32.759 --> 01:09:34.369
from taking the derivative.
01:09:37.130 --> 01:09:38.330
So fine.
01:09:41.370 --> 01:09:43.490
So this is a candidate
equation that
01:09:43.490 --> 01:09:47.420
is obtained in this context--
something that is grown.
01:09:47.420 --> 01:09:49.380
We say we are not
interested in it's
01:09:49.380 --> 01:09:51.850
coming from some
underlying weight,
01:09:51.850 --> 01:09:55.060
but presumably,
this system still,
01:09:55.060 --> 01:09:58.270
if I look at it at
long times, will
01:09:58.270 --> 01:10:01.450
have some kind of fluctuations.
01:10:01.450 --> 01:10:04.450
All the fluctuations of
this growing surface,
01:10:04.450 --> 01:10:07.340
like the fluctuations
of the soap bubble,
01:10:07.340 --> 01:10:10.986
and they have this
logarithmic dependence.
01:10:10.986 --> 01:10:13.194
You have a question?
01:10:13.194 --> 01:10:17.400
AUDIENCE: So why doesn't that
term-- what if I put in h times
01:10:17.400 --> 01:10:19.968
that term that we want
to appear and then I
01:10:19.968 --> 01:10:21.200
vary with respect to h?
01:10:21.200 --> 01:10:24.019
A term like what we want to pop
out together with other terms?
01:10:24.019 --> 01:10:25.810
MEHRAN KARDAR: Yeah,
but those other terms,
01:10:25.810 --> 01:10:27.660
what do you want
to do with them?
01:10:27.660 --> 01:10:32.700
AUDIENCE: Well, maybe they're
not acceptable [INAUDIBLE]?
01:10:32.700 --> 01:10:35.360
MEHRAN KARDAR: So you're
saying why not have a term that
01:10:35.360 --> 01:10:38.500
is h gradient of h squared?
01:10:38.500 --> 01:10:42.690
Functional derivative of that
is gradient of h squared.
01:10:42.690 --> 01:10:47.390
And then you have a term
that is h Laplacian-- it's
01:10:47.390 --> 01:10:50.785
a gradient of,
sorry, gradient of h.
01:10:57.816 --> 01:11:00.690
And then you expand this.
01:11:00.690 --> 01:11:02.820
Among the terms that
you would generate
01:11:02.820 --> 01:11:07.560
would be a term that
is h, a Laplacian of h.
01:11:07.560 --> 01:11:09.745
This term violates
this condition
01:11:09.745 --> 01:11:13.490
that we had over here.
01:11:13.490 --> 01:11:18.000
And you cannot separate
this term from that term.
01:11:18.000 --> 01:11:20.960
So what you
describe, you already
01:11:20.960 --> 01:11:22.940
see at the level over here.
01:11:22.940 --> 01:11:27.380
It violates translation of
symmetry in [? nature ?].
01:11:27.380 --> 01:11:29.775
And you can play around
with other functions.
01:11:29.775 --> 01:11:31.488
You come to the same conclusion.
01:11:35.300 --> 01:11:39.660
OK, so the question is, well,
you added some term here.
01:11:39.660 --> 01:11:43.750
If I look at this surface
that has grown at large time,
01:11:43.750 --> 01:11:46.720
does it have the same
fluctuations as we had before?
01:11:49.330 --> 01:11:54.820
So a simple way
to ascertain that
01:11:54.820 --> 01:11:58.780
is to do the same kind
of dimensional analysis
01:11:58.780 --> 01:12:02.570
which, for the Landau-Ginzburg,
was a prelude to doing
01:12:02.570 --> 01:12:04.810
renormalization.
01:12:04.810 --> 01:12:08.550
So we did things like
epsilon expansion, et cetera.
01:12:08.550 --> 01:12:12.960
But to calculate that there
was a critical dimension of 4,
01:12:12.960 --> 01:12:16.300
all we needed to do
was to rescale x and m,
01:12:16.300 --> 01:12:18.480
and we would
immediately see that mu
01:12:18.480 --> 01:12:21.770
goes to mu, b to the 4
minus D or something--
01:12:21.770 --> 01:12:24.760
D minus 4, for example.
01:12:24.760 --> 01:12:28.970
So we can do the
same thing here.
01:12:28.970 --> 01:12:33.130
We can always move
to a frame that
01:12:33.130 --> 01:12:35.250
is moving with the
average velocity,
01:12:35.250 --> 01:12:37.910
so that we are focusing
on the fluctuations.
01:12:37.910 --> 01:12:40.570
So we can basically
ignore this term.
01:12:40.570 --> 01:12:45.620
I'm going to rescale
x by a factor of b.
01:12:45.620 --> 01:12:50.980
I'm going to rescale time by
a factor of b to something
01:12:50.980 --> 01:12:52.290
to the z.
01:12:52.290 --> 01:12:57.360
And this z is kind
of indicative of what
01:12:57.360 --> 01:13:02.430
we've seen before-- that somehow
in these dynamical phenomena,
01:13:02.430 --> 01:13:08.620
the scaling of time and space
are related to some exponent.
01:13:08.620 --> 01:13:10.690
But there's also
an exponent that
01:13:10.690 --> 01:13:14.690
characterizes how
the fluctuations in h
01:13:14.690 --> 01:13:18.070
grow if I look at systems
that are larger and larger.
01:13:18.070 --> 01:13:20.990
In particular, if I had
solved that equation,
01:13:20.990 --> 01:13:24.210
rather than for a soap
bubble in two dimensions,
01:13:24.210 --> 01:13:28.650
for a line-- for a string that
I was pulling so that I had
01:13:28.650 --> 01:13:31.280
line tension-- the
one-dimensional version of it,
01:13:31.280 --> 01:13:35.860
the one-dimensional version of
an integral of 1 over q squared
01:13:35.860 --> 01:13:37.940
would be something
that would grow
01:13:37.940 --> 01:13:40.100
with the size of the system.
01:13:40.100 --> 01:13:43.100
So there I would have a
chi of 1/2, for example,
01:13:43.100 --> 01:13:44.850
in one dimension.
01:13:44.850 --> 01:13:47.250
So this is the general thing.
01:13:47.250 --> 01:13:51.800
And then I would say that
the first equation, dhy dt,
01:13:51.800 --> 01:13:54.290
gets a factor of
b to the chi minus
01:13:54.290 --> 01:13:58.930
z, because h scaled
by a factor of chi,
01:13:58.930 --> 01:14:06.280
t scaled by a factor of z,
the term sigma Laplacian of h
01:14:06.280 --> 01:14:10.090
gets a factor of b
to the chi minus 2
01:14:10.090 --> 01:14:13.350
from the two derivatives here--
sorry, the z and the 2 look
01:14:13.350 --> 01:14:15.380
kind of the same.
01:14:15.380 --> 01:14:17.412
This is a z.
01:14:17.412 --> 01:14:20.640
This is a 2.
01:14:20.640 --> 01:14:23.830
And then the term
that is proportional
01:14:23.830 --> 01:14:27.740
to this non-linearity that
I wrote down-- actually, it
01:14:27.740 --> 01:14:32.270
is very easy, maybe worthwhile,
to show that sigma to the 4th
01:14:32.270 --> 01:14:36.290
goes with a factor of
b to the chi minus 4.
01:14:36.290 --> 01:14:41.100
It is always down by a factor of
two scalings in b with respect
01:14:41.100 --> 01:14:44.470
to a Laplacian-- the
same reason that when
01:14:44.470 --> 01:14:47.360
we were doing the
Landau-Ginzburg.
01:14:47.360 --> 01:14:50.790
We could terminate the series
at order of gradient squared,
01:14:50.790 --> 01:14:53.610
because higher-order
derivatives were irrelevant.
01:14:53.610 --> 01:14:55.790
They were scaling to 0.
01:14:55.790 --> 01:15:02.890
But this term grows
like b to the 2 chi,
01:15:02.890 --> 01:15:08.820
because it's h squared, minus 2,
because there's two gradients.
01:15:08.820 --> 01:15:11.950
Now thinking about
the scaling of eta
01:15:11.950 --> 01:15:17.670
takes a little bit of thought,
because what we have-- we
01:15:17.670 --> 01:15:21.330
said that the average
of eta goes to 0.
01:15:21.330 --> 01:15:26.290
The average of eta at
two different locations
01:15:26.290 --> 01:15:29.380
and two different times--
it is these particles that
01:15:29.380 --> 01:15:32.460
are raining down--
they're uncorrelated
01:15:32.460 --> 01:15:33.800
at different times.
01:15:33.800 --> 01:15:36.026
They're uncorrelated
at different positions.
01:15:38.600 --> 01:15:40.190
There's some kind
of variance here,
01:15:40.190 --> 01:15:43.110
but that's not important to us.
01:15:43.110 --> 01:15:48.840
If I rescale t by a factor
of b, delta of bt-- sorry,
01:15:48.840 --> 01:15:54.230
if I scale t by a factor of b
to the z, delta of b to the zx
01:15:54.230 --> 01:15:57.550
will get a factor
of b to the minus z.
01:15:57.550 --> 01:16:02.110
This will get a factor
of b to the minus d.
01:16:02.110 --> 01:16:06.240
But the noise, eta,
is half of that.
01:16:06.240 --> 01:16:14.210
So what I will have is b to
the minus z plus d over 2 times
01:16:14.210 --> 01:16:18.180
eta on the rescalings
that I have indicated.
01:16:18.180 --> 01:16:23.780
I get rid of this term.
01:16:23.780 --> 01:16:28.960
So this-- divide by
b to the chi minus z.
01:16:28.960 --> 01:16:37.760
So then this becomes bh y dt
is sigma b to the z minus 2.
01:16:37.760 --> 01:16:44.550
Maybe I'll write it in
red-- b to the z minus 2.
01:16:44.550 --> 01:16:52.470
This becomes sigma to the
4, b to the z minus 4.
01:16:52.470 --> 01:16:54.850
And then lambda over 2.
01:16:54.850 --> 01:16:57.450
This is Laplacian of h.
01:16:57.450 --> 01:17:02.580
This, for derivative of
h, this is Laplacian of h.
01:17:02.580 --> 01:17:09.900
This term becomes b to the chi
plus z minus 2, gradient of h
01:17:09.900 --> 01:17:12.290
squared.
01:17:12.290 --> 01:17:24.804
And the final term becomes b to
the chi minus d minus z over 2.
01:17:36.005 --> 01:17:38.950
AUDIENCE: [INAUDIBLE]?
01:17:38.950 --> 01:17:42.990
MEHRAN KARDAR: b to the
minus chi-- you're right.
01:17:42.990 --> 01:17:51.424
And then, actually, this I-- no?
01:17:54.370 --> 01:18:09.944
Minus chi minus d over 2
minus d over 2 [INAUDIBLE],
01:18:09.944 --> 01:18:10.942
That's fine.
01:18:14.940 --> 01:18:21.011
So I can make this
equation to be invariant.
01:18:31.813 --> 01:18:34.970
So I want to find
out what happens
01:18:34.970 --> 01:18:38.180
to this system if I find
some kind of an equation,
01:18:38.180 --> 01:18:41.155
or some kind of behavior
that is scale invariant.
01:18:41.155 --> 01:18:45.690
You can see that immediately,
my choice for the first term
01:18:45.690 --> 01:18:49.730
has to be z equals to 2.
01:18:49.730 --> 01:18:52.100
So basically, it
says that as long
01:18:52.100 --> 01:18:57.060
as you're governed by
something that is diffusive,
01:18:57.060 --> 01:19:00.700
so that when you go to Fourier
space, you have q squared,
01:19:00.700 --> 01:19:03.430
your relaxation times
are going to have
01:19:03.430 --> 01:19:07.140
this diffusive character,
where time is distance squared.
01:19:07.140 --> 01:19:10.480
Actually, you can see that
immediately from the equation
01:19:10.480 --> 01:19:13.700
that this diffusion time
goes like distance squared.
01:19:13.700 --> 01:19:17.380
So this is just a
statement of that.
01:19:17.380 --> 01:19:21.940
Now, it is the noise that
causes the fluctuations.
01:19:21.940 --> 01:19:29.830
And if I haven't made
some simple error,
01:19:29.830 --> 01:19:33.600
you will find that the
coefficient of the noise term
01:19:33.600 --> 01:19:36.980
becomes scale invariant,
provided that I choose it
01:19:36.980 --> 01:19:41.910
to be z minus d over 2 for chi.
01:19:41.910 --> 01:19:45.610
And since my z was
2, I'm forced to have
01:19:45.610 --> 01:19:48.490
chi to be 2 minus d over 2.
01:19:48.490 --> 01:19:51.770
And let's see if it
makes sense to us.
01:19:51.770 --> 01:20:00.170
So if I have a surface such
as the case of the soap bubble
01:20:00.170 --> 01:20:04.210
in two dimensions, chi is 0.
01:20:04.210 --> 01:20:06.880
And 0 is actually
this limiting case
01:20:06.880 --> 01:20:10.020
that would also be a logarithm.
01:20:10.020 --> 01:20:14.140
If I go to the case of d equals
to 1-- like pulling a line
01:20:14.140 --> 01:20:18.090
and having the line fluctuate--
then I have 2 minus 1
01:20:18.090 --> 01:20:20.550
over 2, which is
1/2, which means
01:20:20.550 --> 01:20:22.700
that, because of
thermal fluctuations,
01:20:22.700 --> 01:20:26.290
this line will look
like it a random walk.
01:20:26.290 --> 01:20:27.710
You go a distance x.
01:20:27.710 --> 01:20:30.420
The fluctuations
in height will go
01:20:30.420 --> 01:20:33.290
like the square root of that.
01:20:33.290 --> 01:20:35.000
OK?
01:20:35.000 --> 01:20:37.220
So all of that is fine.
01:20:37.220 --> 01:20:39.660
You would have done
exactly the same answer
01:20:39.660 --> 01:20:44.210
if you had just gotten a
kind of scaling such as this
01:20:44.210 --> 01:20:46.850
for the case of
the Gaussian Model
01:20:46.850 --> 01:20:49.310
without the nonlinearities.
01:20:49.310 --> 01:20:51.850
But for the Gaussian
Model with nonlinearities,
01:20:51.850 --> 01:20:55.460
we could also then estimate
whether the nonlinearity u
01:20:55.460 --> 01:20:57.210
is relevant.
01:20:57.210 --> 01:21:00.470
So here we see that the
coefficient of our nonlinearity
01:21:00.470 --> 01:21:07.690
is lambda, is
governed by something
01:21:07.690 --> 01:21:11.260
that is chi plus z minus 2.
01:21:11.260 --> 01:21:17.270
And our chi is 2 minus z over 2.
01:21:17.270 --> 01:21:19.720
z minus 2 is 0.
01:21:19.720 --> 01:21:25.720
So whether or not this
nonlinearity is relevant,
01:21:25.720 --> 01:21:28.590
we can see depends
on whether you're
01:21:28.590 --> 01:21:31.940
above or below two dimensions.
01:21:31.940 --> 01:21:35.900
So when you are
below two dimensions,
01:21:35.900 --> 01:21:39.840
this nonlinearity is relevant.
01:21:39.840 --> 01:21:42.720
And you will certainly
have different types
01:21:42.720 --> 01:21:45.810
of scaling phenomena
then what you
01:21:45.810 --> 01:21:52.620
predict by the case of the
diffusion equation plus noise.
01:21:52.620 --> 01:21:56.300
Of course, the
interesting case is
01:21:56.300 --> 01:22:02.120
when you are at the
marginal dimension of 2.
01:22:05.840 --> 01:22:12.030
Now, in terms of when you do
proper renormalization group
01:22:12.030 --> 01:22:15.820
with this nonlinearity,
you will find that,
01:22:15.820 --> 01:22:20.350
unlike the nonlinearity of
the Landau-Ginzburg, which
01:22:20.350 --> 01:22:23.800
is marginally irrelevant
in four dimensions.
01:22:23.800 --> 01:22:29.250
du by dl was minus u
squared, this lambda
01:22:29.250 --> 01:22:31.670
is marginally relevant.
01:22:31.670 --> 01:22:36.320
d lambda by dl is proportional
to plus lambda squared.
01:22:36.320 --> 01:22:44.040
It is relevant marginality--
marginally relevant.
01:22:44.040 --> 01:22:46.340
And actually, the
epsilon expansion
01:22:46.340 --> 01:22:51.250
gives you no information about
what's happening in the system.
01:22:51.250 --> 01:22:55.050
So people have then done
numerical simulations.
01:22:55.050 --> 01:22:57.845
And they find that there
is a roughness that
01:22:57.845 --> 01:23:02.640
is characterized by an exponent,
say something like 0.4.
01:23:02.640 --> 01:23:06.780
So that when you look at
some surface that is grown,
01:23:06.780 --> 01:23:12.130
is much, much rougher than
the surface of a soap bubble
01:23:12.130 --> 01:23:15.950
or what's happening on
the surface of the pond.
01:23:15.950 --> 01:23:21.820
And the key to all of this is
that we wrote down equations
01:23:21.820 --> 01:23:24.410
on the basis of
this generalization
01:23:24.410 --> 01:23:27.760
on symmetry that we had
learned, now applied
01:23:27.760 --> 01:23:31.310
to this dynamical
system, did an expansion,
01:23:31.310 --> 01:23:34.640
found one first term.
01:23:34.640 --> 01:23:37.330
And we found it to be relevant.
01:23:37.330 --> 01:23:39.980
And is actually not
that often that you
01:23:39.980 --> 01:23:42.460
find something that is
relevant, because then it
01:23:42.460 --> 01:23:44.990
is a reason to celebrate.
01:23:44.990 --> 01:23:47.150
Because most of the time,
things are irrelevant,
01:23:47.150 --> 01:23:50.260
and you end up with boring
diffusion equations.
01:23:50.260 --> 01:23:53.170
So find something
that is relevant.
01:23:53.170 --> 01:23:56.010
And that's my last
message to you.